| L(s) = 1 | + 2·2-s + 3·4-s + 2·5-s + 4·8-s + 4·10-s + 2·11-s + 4·13-s + 5·16-s − 2·17-s − 4·19-s + 6·20-s + 4·22-s + 8·23-s + 8·26-s + 10·29-s + 4·31-s + 6·32-s − 4·34-s − 8·37-s − 8·38-s + 8·40-s − 6·41-s + 10·43-s + 6·44-s + 16·46-s + 6·47-s + 12·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.41·8-s + 1.26·10-s + 0.603·11-s + 1.10·13-s + 5/4·16-s − 0.485·17-s − 0.917·19-s + 1.34·20-s + 0.852·22-s + 1.66·23-s + 1.56·26-s + 1.85·29-s + 0.718·31-s + 1.06·32-s − 0.685·34-s − 1.31·37-s − 1.29·38-s + 1.26·40-s − 0.937·41-s + 1.52·43-s + 0.904·44-s + 2.35·46-s + 0.875·47-s + 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.62188875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.62188875\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 28 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 76 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 59 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 22 T + 232 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20 T + 215 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 115 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 26 T + 304 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.904405456122294738866332507150, −8.783201829758983090886125222924, −8.287876696519847978081697606829, −7.87373762664283195166597787366, −7.38019787192156454827497270862, −6.71166928432622107145834167435, −6.57956568922786304548301987761, −6.52942671909187957809797676204, −5.83944179794273176742325707652, −5.66966395975908589821427917253, −5.02819770508673162251060996430, −4.81783855596281338002717693695, −4.17757073131082453283400669081, −4.11059276021954986481199305777, −3.19232177644999562790647870256, −3.18870127220997976273791648814, −2.44581985130163910558282353412, −2.04676595535829612878936754493, −1.39005477331179646388410122653, −0.895046676809971923298890429254,
0.895046676809971923298890429254, 1.39005477331179646388410122653, 2.04676595535829612878936754493, 2.44581985130163910558282353412, 3.18870127220997976273791648814, 3.19232177644999562790647870256, 4.11059276021954986481199305777, 4.17757073131082453283400669081, 4.81783855596281338002717693695, 5.02819770508673162251060996430, 5.66966395975908589821427917253, 5.83944179794273176742325707652, 6.52942671909187957809797676204, 6.57956568922786304548301987761, 6.71166928432622107145834167435, 7.38019787192156454827497270862, 7.87373762664283195166597787366, 8.287876696519847978081697606829, 8.783201829758983090886125222924, 8.904405456122294738866332507150
